Where are you located in OH?
I've caught cats on public and private land, however when I set on public land I'm always sure to set so far away from a parking area or any established trails that the public may use for walking to minimize the chances of catching a domestic that someone has out for a walk... lets not even begin to discuss leash laws or controlling you domestic dog off-leash on public land, lol...
PA has a pretty good cat population, but it's nothing like other states where you can catch dozens of cats in a few weeks. That being said, it will require some luck and/or research on your behalf to get onto some cats. There are some places in PA that seem to have much higher bobcat populations than others, so with a little research you can probably find catch numbers by WMU and make your deductions from there.
Permissions haven't been that difficult for me to obtain, generally speaking. A lot of landowners want predators and nuisance animals trapped off their land, but they don't want their deer hunting affected. If I'm approaching a landowner that uses his land for hunting whitetail, I ask if it's archery, rifle, or both. Generally rifle hunters won't mind letting you on to trap coyotes at the beginning of our furbearer season. You're after a bobcat, though, so that season is after our firearms season for deer- so that helps your chances of gaining permission. I think the hardest part for you will be finding quality trapping land that isn't being trapped by someone else, but public land is fair game.... When I see where you are in OH I can probably make some recommendations on where to start looking. Do you have OnX?
What a genuine response. I've thought about asking the same questions, but didn't think I'd get much feedback.
Is trapping pretty open on state land in PA? From my experience, basically only water trapping is allowed on state properties in Indiana. They all vary here, but every one I've asked only allows water trapping (except for DPs and cages). Too much concern with domestics, but they all want help controlling beaver and coon. The national forest is different.